Conservation of mechanical energy clarification of explanation in textbook

Question

I have a bit of a problem with the explanation of conservation of mechanical energy in the open source textbook "College Physics" by Openstax.

The problem stems from their use of the term the system: each time this phrase is used I suspect it does not refer to the same system. This is fine for people that already know this stuff, but it is an introductory book and I think it should be made explicit for the level of the readers.

The section under consideration is the following:

Conservation of Mechanical Energy
Let us now consider what form the work-energy theorem takes when only conservative forces are involved. This will lead us to the conservation of energy principle. The work-energy theorem states that the net work done by all forces acting on a system equals its change in kinetic energy. In equation form, this is \begin{equation}\label{eq:743} W_{\text{net}} = \frac{1}{2} m v^2 - \frac{1}{2} m v_0^2 = \Delta \text{KE}. \tag{7.43} \end{equation} If only conservative forces act, then \begin{equation}\label{eq:744} W_{\text{net}} = W_c, \tag{7.44} \end{equation} where $W_c$ is the total work done by all conservative forces. Thus, \begin{equation}\label{eq:745} W_c = \Delta \text{KE}. \tag{7.45} \end{equation} Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, $W_c = - \Delta \text{PE}$. Therefore, \begin{equation}\label{eq:746} - \Delta \text{PE} = \Delta \text{KE} \tag{7.46} \end{equation} or \begin{equation}\label{eq:747} \Delta \text{KE} + \Delta \text{PE} = 0. \tag{7.47} \end{equation} This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces.

My analysis:

we will analyse the above as though the conservative force was gravity, and it has its effect on an object in the Earth-object system.

Conservation of mechanical energy has to be concerned with conservation of energy within the Earth-object system, since it involves $\text{PE}$.

\eqref{eq:743} is said to be concerned with net work done by forces acting on a system, but what system will be the one we will be concerned with?

\eqref{eq:744} relates to conservative forces, the conservative force of gravity acts within the Earth-object system.

Therefore, the system that this conservative force acts on has to be the object in the Earth-object system.

Therefore \eqref{eq:745} refers to the kinetic energy of the object in the Earth-object system.

We therefore have problem, as the result \eqref{eq:747} has to be concerned with Earth-object system as it contains $\text{PE}$.

My attempt at resolving this.

The conservative force of gravity acts on the object to increase its $\text{KE}$. The $\text{KE}$ of the Earth-object system is the sum of the $\text{KE}$ of everything in it (I am guessing this is how things work, as it's the only way I can make sense of this, is this correct?) so when the object's kinetic energy increases, the $\text{KE}$ of the Earth-object system increases by the same amount.

Have I reasoned this out correctly? If not, what am I missing?

Answer 1

7.43 is said to be concerned with net work done by forces acting on a system, but what system will be the one we will be concerned with?

The Earth-object system.

The force of gravity is force internal to the system. In this case an isolated system. For an isolated system (one not subjected to a net external force) total mechanical energy (KE + PE) is conserved. There is, however, a caveat. None of the internal forces can be dissipative (non conservative), such as friction. For a falling object that means mechanical energy is only conserved if there was no air drag, a form of friction. Then 7.45 is applicable.

Therefore 7.45 refers to the kinetic energy of the object in the earth object system.

Actually, it applies to both the object and the Earth. The Earth pulls down on the object and the object pulls up on the Earth per Newton's 3rd law conserving system momentum. But for objects of mass much less than the Earth, the change in kinetic energy of the Earth would be infinitesimal.

We therefore have problem as the result 7.47 has to be concerned with e-o system as it contains PE

Not if you realize that the change in KE in 7.45 and 7.46 applies to the Earth as well as the object, of the Earth-Object system, as I previously stated. Then the total decrease in PE of the Earth-Object system equals the gain in KE of the Earth-Object system.

Hope this helps.

Answer 2

I disagree with the accepted answer, even though the physics presented in it is correct.

The problem is more one of pedagogy. A lot of questions on PSE come from what I call "a failure to abstract".

In the context of the book, there is no Earth. "The System" is a universe where there is a constant uniform gravitational field, with some perfectly elastic masses and ideal springs applying forces on each other.

Based on The System's symmetries, we can expect certain things to be conserved.

It doesn't change with time, so energy is conserved (provided you include a potential energy per unit mass:

$$ U(x, y, z) = gz $$

It's also invariant under $x$ and $y$ translations, so horizontal momentum is conserved.

Since $U(z+dz) \ne U(z)$, The System is not invariant under vertical translations, so $p_z$ is not conserved.